Updated · Built and checked by the PlinzoTools team, formulas tested against published sources
Calculate Pearson's correlation coefficient (r) between two data sets.
Positive correlation
Pearson's correlation coefficient (r) measures the strength and direction of the linear relationship between two paired data sets, ranging from -1 (perfect negative correlation) to +1 (perfect positive correlation).
Enter two equal-length data sets (X values and Y values, paired by position). Pearson's r appears instantly.
r = Σ(x - x̄)(y - ȳ) / √(Σ(x - x̄)² × Σ(y - ȳ)²)
|r| ≥ 0.8 is generally considered a strong correlation, 0.5-0.8 moderate, 0.2-0.5 weak, and below 0.2 very weak or none. These thresholds are rules of thumb, not universal cutoffs -- context matters.
The tool loads with two small paired lists: X = 1, 2, 3, 4, 5 and Y = 2, 4, 5, 4, 5. These are hypothetical values, and they make a good example to follow by hand.
r = 6 / √(10 × 6) = 6 / √60 ≈ 0.7746
By the page's rule-of-thumb bands, that is a moderate to strong positive correlation, and the tool labels it "Moderate" because the value is below 0.8.
Squaring r gives the proportion of variation in one variable that a straight-line relationship with the other accounts for. Here, 0.7746 squared is about 0.60, so roughly 60 percent of the variation in Y is linearly associated with X in this small example. The tool reports r itself, so you can square it yourself if you need this figure. Keep in mind that this describes the fit to the data you entered and does not prove that one variable drives the other.
A positive r means the two variables tend to rise together, and a negative r means one tends to rise as the other falls. The size shows how tightly the points follow a straight line, not how steep the line is. Two data sets can have the same r while one line rises gently and the other steeply. If you need the slope or an actual prediction, you need a regression line, which this tool does not compute.
No -- a strong correlation only shows the two variables move together, not that one causes the other. Confounding variables can produce strong correlations between unrelated things.
No linear relationship -- though a non-linear relationship (e.g. a U-shape) could still exist even with r near 0, since Pearson's r only captures linear association.
It shows a message asking for two equal-length data sets with at least two pairs each, and no correlation is calculated.
Not necessarily. A high r shows a tight linear pattern in your data. Whether it matters depends on the context, the sample size, and whether a real cause exists.
Multiply the displayed r by itself. For example, r = 0.7746 gives r squared of about 0.60.
Pearson's r only detects linear relationships. A U-shaped or other curved pattern can produce an r close to zero.
Reference: NIST Weights and Measures